Abstract:
I discuss Schwinger's treatment of quantum systems of a finite number of degrees of freedom.  Starting from a normal operator with a finite discrete spectrum he constructs a pairs of complementary unitary operators that are finite dimensional versions of Weyl pairs.These pairs are then decomposed into tensor products of pairs with prime numbers of eigenvalues.  This gives a general decomposition of of general finite quantum system into irreducible unitary building blocks.  This leads to the question of how to represent these Weyl pairs acting on prime dimensional spaces using quantum algorithms with q-bits?